Critical Points & Second Derivative Test - Maple Help

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Critical Points and the Second Derivative Test

 Description Determine and classify the critical points of a multivariate function.

Critical Points and the Second Derivative Test

Objective Function $f$

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 ${{x}}^{{2}}{+}{3}{}{x}{}{{y}}^{{2}}{-}{5}{}{{y}}^{{3}}$ (1)

List of Independent Variables

 > $v≔\left[{x}{,}{y}\right]$
 ${v}{:=}\left[{x}{,}{y}\right]$ (2)

Equations $\nabla f=\mathbf{0}$

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 $\left[{2}{}{x}{+}{3}{}{{y}}^{{2}}{,}{6}{}{x}{}{y}{-}{15}{}{{y}}^{{2}}\right]$ (3)

Critical Points

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 $\left[\left[{0}{,}{0}\right]{,}\left[{-}\frac{{25}}{{6}}{,}{-}\frac{{5}}{{3}}\right]\right]$ (4)

Second Derivative Test

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{SecondDerivativeTest}\right]\left(,v=\right);$
 ${\mathrm{LocalMin}}{=}\left[{}\right]{,}{\mathrm{LocalMax}}{=}\left[{}\right]{,}{\mathrm{Saddle}}{=}\left[\left[{-}\frac{{25}}{{6}}{,}{-}\frac{{5}}{{3}}\right]\right]$ (5)

Hessians and their Eigenvalues

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 $\left[\begin{array}{rr}{2}& {0}\\ {0}& {0}\end{array}\right]{,}\left[{0}{,}{2}\right]$
 $\left[\begin{array}{rr}{2}& {-}{10}\\ {-}{10}& {25}\end{array}\right]{,}\left[\frac{{27}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{929}}{,}\frac{{27}}{{2}}{-}\frac{{1}}{{2}}{}\sqrt{{929}}\right]$ (6)